Shawn Zhong

Shawn Zhong

钟万祥
  • Tutorials
  • Mathematics
    • Math 240
    • Math 375
    • Math 431
    • Math 514
    • Math 521
    • Math 541
    • Math 632
    • Abstract Algebra
    • Linear Algebra
    • Category Theory
  • Computer Sciences
    • CS/ECE 252
    • CS/ECE 352
    • Learn Haskell
  • Projects
    • 2048 Game
    • HiMCM 2016
    • 登峰杯 MCM
  • Course Notes
    • AP Microecon
    • AP Macroecon
    • AP Statistics
    • AP Chemistry
    • AP Physics E&M
    • AP Physics Mech
    • CLEP Psycho

Shawn Zhong

钟万祥
  • Tutorials
  • Mathematics
    • Math 240
    • Math 375
    • Math 431
    • Math 514
    • Math 521
    • Math 541
    • Math 632
    • Abstract Algebra
    • Linear Algebra
    • Category Theory
  • Computer Sciences
    • CS/ECE 252
    • CS/ECE 352
    • Learn Haskell
  • Projects
    • 2048 Game
    • HiMCM 2016
    • 登峰杯 MCM
  • Course Notes
    • AP Microecon
    • AP Macroecon
    • AP Statistics
    • AP Chemistry
    • AP Physics E&M
    • AP Physics Mech
    • CLEP Psycho

Category Theory

Home / Mathematics / Notes / Category Theory

范畴论 Category Theory

  • May 23, 2018
  • Shawn
  • Category Theory
  • No comments yet

Category Theory: The Beginner’s Introduction

Playlist: https://www.youtube.com/playlist?list=PLm_IBvOSjN4zthQSQ_Xt6gyZJZZAPoQ6v
Video 1 Notes https://youtu.be/P6DvIfTJhx8
Video 2 Notes https://youtu.be/4fggq5U3Khg
Video 3 Notes https://youtu.be/hYO14y50Uso
Video 4 Notes https://youtu.be/80bbALpA8k8
Video 5 Notes https://youtu.be/Z-qL2G60zJk
Video 6 Notes https://youtu.be/TsogM9CEbUk
Read More >>

Category Theory - Video 1

  • May 23, 2018
  • Shawn
  • Category Theory
  • No comments yet
First Section Will Cover • Category of Sets • Sets with a Endomap • Category of Permutations Structure of This Course • Category Theory Overview • Category Theory Deep-Dive • Music Theory Formulation • Implement Theory in Software Topics Terminal Object Equalizer Initial Object Co-Equalizer Monomorphism Product Epimorphism Co-Product Pullback Sub-Object Pushout Map Object Categories Variable Sets Posets Dynamical Systems Bouquets Permeations Pointed Sets Graphs Bisets The 12 Pitch-Classes in Music • There are groups of 2 black notes (in green), and groups of 3 (in purple) • The note C is denoted in orange • Major in C • There are two names for the black note • In some cases one name must be used instead of the other
Read More >>

Category Theory - Video 2

  • May 23, 2018
  • Shawn
  • Category Theory
  • No comments yet
Definition of Category • Data General S (Set) A set of Objects of the Category In S, the Objects are Abstract Sets e.g. A,B,C,X_i, etc A set of Arrows between Objects In S, the arrows are functions The source is called the Domain (also called morphisms or maps) The target is called the Codomain ∀a∈A, ∃!b∈B s.t. f(a)=b e.g. f:A→B A special arrow called The Identity Arrow In S, this is the identity map: defined on each object in the Category ∀a∈A,1_A (a)=a 1_A:A→A • Rules ○ Compositions § Given two arbitrary arrows A →┴f B_1, and B_2 →┴g C § We can form the composite A→┴(g∘f) C (called g following f) iff B_1=B_2 § e.g. in the case where we have A→┴f B→┴g C ○ Associative § Compositions of arbitrary arrows A →┴f B, B →┴g C, and C→┴hD is associative § i.e. the following relations holds: h∘(g∘f)=(h∘g)∘f ○ Identity Laws § For arbitrary objects A and B § The arrows A→┴1_A A, B→┴1_B B, and A→┴f B must obey the following laws § f∘1_A=1_B∘f=f Objects • Objects in S ○ The Objects in S are Abstract Sets ○ We will represent them, for example, as: A,B,C,X_i,Y_j^′ • Elements in Abstract Sets ○ If we have a set A of size 2, and a set B of size 3, for example ○ We will use the following notation to refer to the unique
Read More >>

Category Theory - Video 3

  • May 23, 2018
  • Shawn
  • Category Theory
  • No comments yet
Category Music (S) • Objects ○ Pitch Classes § X={x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_10,x_11 } ○ Pitch-Class Names § Z={C♮,C♯,D♭,D♮,D♯,E♭,E♮,F♮,F♯,G♭,G♮,G♯,A♭,A♮,A♯,B♭,B♮} ○ Letter Names of the Pitch-Class Names § L={C,D,E,F,G,A,B} • Arrows ○ n:Z→X § Assign to each name its pitch class ○ t:Z→L § Give the letter name of each pitch-class name ○ i:L→Z § Represents the Major Mode ○ j:L→Z § Represents the Minor Mode Defining Composition • Recall the composition part in the definition of Category ○ Given two arbitrary arrows A →┴f B_1, and B_2 →┴g C ○ We can form the composite A→┴(g∘f) C (called g following f) iff B_1=B_2 ○ e.g. in the case where we have A→┴f B→┴g C • We can rename the composite r and redraw the diagram • But we have to state r=g∘f, because there could be many arrows A→C • To indicate that r=g∘f, we can simply say that this diagram commutes • A Commutative Diagram in any Category is one in which all paths between two objects must be interpreted as the same arrow Composition: Abstract Example • Let s say we have the following three Sets A,B and C ○ A={a_0,a_1 } ○ B={b_0,b_1,b_2 } ○ C={c_0,c_1 } • And the following two maps ○ f:A→B≡{█(f(a_0 )=b_0@f(a_1 )=b_1 )┤ ○ g:B→C≡{█(g(b_0 )=g(b_1 )=c_0@g(b_2 )=c_1 )┤ • We can form g∘f:A→C (which we renamed r), by first applying f then applying g ○ r(a_0 )=(g∘f)(a_0 )=g(f(a_0 ))=g(b_0 )=c_0 ○ r(a_1 )=(g∘f)(a_1 )=g(f(a_1 ))=g(b_1 )=c_0 Composition in Music (S) • We have maps i:L→Z, j:L→Z, we also have a map n:Z→X. So we have • We have the C Minor Scale as a map j:L→Z and the map t:Z→L, so t∘j:L→L exists • So t∘j:L→L equals 1_L:L→L, because t and j have a special relationship
Read More >>

Category Theory - Video 4

  • May 23, 2018
  • Shawn
  • Category Theory
  • No comments yet
The Identity Arrow • In S, the Identity Arrow takes each element to itself • e.g. for A={a_0,a_1,a_2 },1_A:A→A≡{█(1_A (a_0 )=a_0@1_A (a_1 )=a_1@1_A (a_2 )=a_2 )┤ • In Music (S), we officially have three more arrows ○ 1_X:X→X ○ 1_Z:Z→Z ○ 1_L:L→L The Identity Laws • Suppose we have a category with objects A and B, then we have 1_A:A→A, 1_B:B→B • The Identity Laws can now be restated to say that, in any Category, this Diagram must Commute • For a diagram to commute, all paths between two objects must be interpreted as the same arrow • Example ○ Suppose f(a_0 )=b_0, then ○ (f∘1_A )(a_0 )=f(1_A (a_0 ))=f(a_0 )=b_0 ○ (1_B∘f)(a_0 )=1_B (f(a_0 ))=1_B (b_0 )=b_0 The Category of Sets With an Endomorphism S^↺ • A^(↺1_A ) and B^(↺1_B ) are objects in S^↺, but what shoud the arrows be? • We want every f:A→B in S to be an arrow f:A^(↺1_A )→B^(↺1_B ) in S^↺ • The law f must follow in S: f∘1_A=1_B∘f, can we generalize this? • What if we replaced 1_A with any endomorphism α, and 1_B with β • Then we get a map f:A^(↺1_A )→B^(↺1_B ) in S^↺ that satisfy: f∘α=β∘f The Associative Law • We have four objects, A,B,C, and D, with arrows f:A→B, g:B→C, and h:C→D • The associativity law says that this diagram must commute • We have 3 paths A→D. They are h∘(g∘f),(hg)∘f, and h∘g∘f • They all must be interpreted as the same map A→D • Example ○ A={a_0,a_1 }, B={b_0,b_1,b_2 }, C={c_0,c_1 },D={d_0,d_1,d_2 } ○ f:A→B≡{█(f(a_0 )=b_0@f(a_1 )=b_1 )┤ ○ g:B→C≡{█(g(b_0 )=g(b_2 )=c_1@g(b_1 )=c_0 )┤ ○ h:C→D≡h(c_0 )=h(c_1 )=d_1
Read More >>
  • 1
  • 2

Search

  • Home Page
  • Tutorials
  • Mathematics
    • Math 240 - Discrete Math
    • Math 375 - Linear Algebra
    • Math 431 - Intro to Probability
    • Math 514 - Numerical Analysis
    • Math 521 - Analysis I
    • Math 541 - Abstract Algebra
    • Math 632 - Stochastic Processes
    • Abstract Algebra @ 万门大学
    • Linear Algebra @ 万门大学
    • Category Theory
  • Computer Sciences
    • CS/ECE 252 - Intro to Computer Engr.
    • CS/ECE 352 - Digital System Fund.
    • Learn Haskell
  • Projects
    • 2048 Game
    • HiMCM 2016
    • 登峰杯 MCM
  • Course Notes
    • AP Macroeconomics
    • AP Microeconomics
    • AP Chemistry
    • AP Statistics
    • AP Physics C: E&M
    • AP Physics C: Mechanics
    • CLEP Psychology

WeChat Account

Categories

  • Notes (418)
    • AP (115)
      • AP Macroeconomics (20)
      • AP Microeconomics (23)
      • AP Physics C E&M (25)
      • AP Physics C Mechanics (28)
      • AP Statistics (19)
    • Computer Sciences (2)
    • Mathematics (300)
      • Abstract Algebra (29)
      • Category Theory (7)
      • Linear Algebra (29)
      • Math 240 (42)
      • Math 375 (71)
      • Math 514 (18)
      • Math 521 (39)
      • Math 541 (39)
      • Math 632 (26)
  • Projects (4)
  • Tutorials (11)

Archives

  • October 2019
  • May 2019
  • April 2019
  • March 2019
  • February 2019
  • December 2018
  • November 2018
  • October 2018
  • September 2018
  • July 2018
  • May 2018
  • April 2018
  • March 2018
  • February 2018
  • January 2018
  • December 2017
  • November 2017
  • October 2017
  • September 2017
  • August 2017
  • July 2017
  • June 2017

WeChat Account

Links

RobeZH's thoughts on Algorithms - Ziyi Zhang
Copyright © 2018.      
TOP